Vol. 11, No. 1, 2018

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Continuous dependence and differentiating solutions of a second order boundary value problem with average value condition

Jeffrey W. Lyons, Samantha A. Major and Kaitlyn B. Seabrook

Vol. 11 (2018), No. 1, 95–102
DOI: 10.2140/involve.2018.11.95
Abstract

Using a few conditions, continuous dependence, and a result regarding smoothness of initial conditions, we show that derivatives of solutions to the second order boundary value problem ${y}^{\prime \prime }=f\left(x,y,{y}^{\prime }\right)$, $a, satisfying $y\left({x}_{1}\right)={y}_{1}$, $1∕\left(d-c\right){\int }_{c}^{d}y\left(x\right)\phantom{\rule{0.3em}{0ex}}\underset{̣}{x}={y}_{2}$, where $a<{x}_{1} and ${y}_{1},{y}_{2}\in ℝ$ with respect to each of the boundary data ${x}_{1}$, ${y}_{1}$, ${y}_{2}$, $c$, $d$ solve the associated variational equation with interesting boundary conditions. Of note is the second boundary condition, which is an average value condition.

Keywords
continuous dependence, boundary data smoothness, average value condition, Peano's theorem
Primary: 34B10